Average Error: 34.1 → 8.5
Time: 4.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.34330927986614228 \cdot 10^{53}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -5.2259753180056327 \cdot 10^{-138}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 5.178734871298619 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -8.34330927986614228 \cdot 10^{53}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -5.2259753180056327 \cdot 10^{-138}:\\
\;\;\;\;\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 5.178734871298619 \cdot 10^{102}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r96664 = b;
        double r96665 = -r96664;
        double r96666 = r96664 * r96664;
        double r96667 = 4.0;
        double r96668 = a;
        double r96669 = c;
        double r96670 = r96668 * r96669;
        double r96671 = r96667 * r96670;
        double r96672 = r96666 - r96671;
        double r96673 = sqrt(r96672);
        double r96674 = r96665 - r96673;
        double r96675 = 2.0;
        double r96676 = r96675 * r96668;
        double r96677 = r96674 / r96676;
        return r96677;
}
↓
double f(double a, double b, double c) {
        double r96678 = b;
        double r96679 = -8.343309279866142e+53;
        bool r96680 = r96678 <= r96679;
        double r96681 = -1.0;
        double r96682 = c;
        double r96683 = r96682 / r96678;
        double r96684 = r96681 * r96683;
        double r96685 = -5.225975318005633e-138;
        bool r96686 = r96678 <= r96685;
        double r96687 = 1.0;
        double r96688 = r96678 * r96678;
        double r96689 = 4.0;
        double r96690 = a;
        double r96691 = r96690 * r96682;
        double r96692 = r96689 * r96691;
        double r96693 = r96688 - r96692;
        double r96694 = sqrt(r96693);
        double r96695 = r96694 - r96678;
        double r96696 = sqrt(r96695);
        double r96697 = r96687 / r96696;
        double r96698 = 2.0;
        double r96699 = r96698 * r96690;
        double r96700 = r96692 / r96699;
        double r96701 = r96700 / r96696;
        double r96702 = r96697 * r96701;
        double r96703 = 5.178734871298619e+102;
        bool r96704 = r96678 <= r96703;
        double r96705 = -r96678;
        double r96706 = r96705 - r96694;
        double r96707 = r96706 / r96699;
        double r96708 = 1.0;
        double r96709 = r96678 / r96690;
        double r96710 = r96683 - r96709;
        double r96711 = r96708 * r96710;
        double r96712 = r96704 ? r96707 : r96711;
        double r96713 = r96686 ? r96702 : r96712;
        double r96714 = r96680 ? r96684 : r96713;
        return r96714;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target21.1
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.343309279866142e+53

    1. Initial program 57.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 3.2

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -8.343309279866142e+53 < b < -5.225975318005633e-138

    1. Initial program 37.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv37.6

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]
    4. Using strategy rm

    5. Applied flip--37.6

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{1}{2 \cdot a}\]

    6. Simplified16.0

      \[\leadsto \frac{\color{blue}{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{2 \cdot a}\]

    7. Simplified16.0

      \[\leadsto \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{2 \cdot a}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt16.2

      \[\leadsto \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}} \cdot \frac{1}{2 \cdot a}\]

    10. Applied *-un-lft-identity16.2

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)\right)}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{2 \cdot a}\]

    11. Applied times-frac16.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\right)} \cdot \frac{1}{2 \cdot a}\]

    12. Applied associate-*l*16.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \left(\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{2 \cdot a}\right)}\]

    13. Simplified15.8

      \[\leadsto \frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

    if -5.225975318005633e-138 < b < 5.178734871298619e+102

    1. Initial program 11.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    if 5.178734871298619e+102 < b

    1. Initial program 47.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 3.4

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified3.4

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.34330927986614228 \cdot 10^{53}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -5.2259753180056327 \cdot 10^{-138}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 5.178734871298619 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))